Find the Intersection Using the Midpoint
[color=#000000]The applet below contains a parallelogram graphed in the coordinate plane. [br]Notice how 3 of its vertices have variable coordinates. [br]Note that one vertex is fixed at (0,0). [br][br]You can move the [b]BIG POINTS[/b] anywhere you'd like. [/color]
[color=#000000]Given that 0 has coordinates (0,0), move the points N to (6,0), M to (7,4), and L to (1,4) in the graph above.[/color][b]Move the line segments above the parallelogram using their points to draw [/b][math]\overline{LM}[/math][b] and [/b][math]\overline{OM}[/math][b]. [color=#ff00ff](Remember they were called the "diagonals" of a parallelogram.)[/color] Move point P to their intersection. What do you get as its coordinates? [br][br][/b][b][color=#ff00ff]Write the coordinate values in a pair (e.g., "(2,3).")[/color][/b]
[color=#000000]Recall the theorem we've proven: [/color]
Back to our diagram: How will you use the Parallelogram Diagonals Theorem to find the coordinates of the intersection?[br][br]By the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each[br]other. So, the coordinates of the intersection are the [u][color=#ff00ff](fill-in-the-blank)[/color][/u] of diagonals [math]\overline{LN}[/math] and [math]\overline{OM}[/math].
In the applet above, recall the values of the coordinates of the points [i]O[/i] and [i]M. [/i]Designate them as ([i]x[sub]1[/sub][/i],[i]y[sub]1[/sub][/i]) and ([i]x[sub]2[/sub][/i],[i]y[sub]2[/sub][/i]). I already did it for [i]O[/i]:[br][br][i]O[/i] = ([i]x[sub]1[/sub][/i],[i]y[sub]1[/sub][/i]) = (0,0)[br][i]M[/i] = ([i]x[sub]2[/sub][/i],[i]y[sub]2[/sub][/i]) = [color=#ff00ff](fill-in-the-blank)[/color]
Finally, using the Midpoint Formula, find the midpoint of [i]O [/i]and [i]M[/i].
Does the midpoint you found in Step 4 match where you placed the point [i]P[/i] in the applet above? Why or why not?